Digital filters have been used for a relatively long time for receiver systems in modern NMR and MRI spectrometers. They are generally designed as low-pass filters, and are applied together with the over-sampling and decimation methods. Since the receiver system usually contains a quadrature detector, two digital filters are required for each receiver system.
FIG. 6 shows the last function blocks of a modern NMR receiver system with two quadrature channels A and B including the two digital filters 3a and 3b, and also the two digitizers 2a and 2b for digitizing the two analog NMR signals of the channels A and B.
The sampling rate of the digitizers is normally chosen as large as possible to be more flexible in the design of the analog anti-aliasing filters and to profit from the resolution gain of the over-sampling method. Since, however, the digital filter delimits the frequency bandwidth of the filtered NMR signal and thereby also reduces the maximum frequency which can occur in the filtered NMR signal, much more data is produced than is required by the Nyquist theorem. For this reason, the data rate is additionally reduced after the digital filter (decimation method).
Analog low-pass filters, so-called anti-aliasing filters, are disposed before the two digitizers, whose cut-off frequency must be lower than half the over-sampling rate in order to satisfy the Nyquist criterion. The output signals of the two digital filters are fed into a computer where they are subjected to a calculation process RV and transformed into the desired NMR spectrum or MRI image through subsequent Fourier transformation.
NMR and MRI without digital filters is essentially unfeasible today, since the digital filters have the following important advantages:                1. They have a pass band with a very flat amplitude characteristic which practically does not falsify the signal amplitudes. This is of particular importance for the integration of the NMR signals.        2. The transition band which defines the transition from the pass band to the stop band, may be very narrow, such that a clearly defined region of the NMR spectrum can be cut out.        3. NMR signals within the stop band can be highly suppressed, even if they are very close to the transition region, and can therefore not be folded back into the desired pass band in subsequent decimation.        
Two basic types of digital filters have established themselves among those which are in use, namely the “Infinite Impulse Response” Filter (=IIR filter) and the “Finite Impulse Response” Filter (=FIR filter). NMR and MRI almost exclusively use the exact linear-phased FIR filter, since it produces no phase-related distortions and has a finite pulse response function thereby delimiting the filtered NMR signal in time.
The most important disadvantage of the application of linear-phased FIR filters in NMR and MRI is the shape of the NMR signal in the time domain at the output of the filter. This reflects the influence of the large group delay time TB which is typical for linear-phased and steep digital filters, such that the FID signal appears only after a long and slowly rising oscillation of length TB. This requires very large linear phase correction in the NMR spectrum. Moreover, a long and slowly decaying oscillation, also of length TB, is present at the end of the FID signal.
There are additional further features:                more data points are generated at the filter output than are entered at the input. This feature can be compensated for through suitable measures.        when window functions and/or “backward linear prediction” methods (BLP) are used, the influence of the group delay time must be considered.        the influence of the group delay time must also be taken into consideration for compensating a possible DC portion in the filtered NMR signal.        
FIG. 5, region 2 shows the NMR signal F′ at the output of a linear-phased FIR filter. It is composed of the rising oscillation B1 of a length TB, the actual NMR signal [F]′ of a length TA, and of the decaying oscillation B2 of length TB. The rising and decaying oscillations are composed not only of the portions B1 or B2 but also have portions [B1] or [B2] which are both within [F]′. It should also be noted that the detection time TERF of the NMR signal after the FIR filter is larger that that before the FIR filter, namely (TB+TA+TB) compared to TA.
To simplify representation of the filtered NMR signal including the rising and decaying oscillations, a simplified symbolic illustration in accordance with region 3 of FIG. 5 shall be used below instead of the precise illustration with plotted rising and decaying oscillations in accordance with region 2 of FIG. 5.
As will be apparent below, it is advisable to select the group delay time TB of the digital filter such that exact separation of the three signal regions B1, [F]′ and B2 is possible. This is obtained by selecting the group delay time TB as an integer multiple of the period time of the decimated sampling rate. This condition is not absolutely necessary but considerably facilitates signal processing.
In NMR and also in MRI, the final aim is not the NMR signal (=time signal) but the NMR spectrum or the MRI image. Both require that the NMR signal be Fourier transformed, i.e. be transformed from the time to the frequency domain. To prevent falsification or additional distortions in the NMR spectrum with this transformation, the NMR spectra of the filtered and non-filtered NMR signals should coincide to an optimum extent except for the filter function. This goal is obtained by initially carrying out a special calculation process RV on the filtered NMR signal before it is Fourier transformed.
The calculation process RV reduces the detection time (TB+TA+TB) of the filtered NMR signal F′ down to the detection time TA of the NMR signal F at the filter input. Thereby the calculation process has to prevent the generation of any additional distortions in the frequency spectrum as far as possible. In this manner, the condition can be met that the NMR spectra of the filtered and non-filtered NMR signals maximally correspond to each other except for the filter function.
The correctness of the above-mentioned procedure is obvious by considering that the Fourier transformation implicitly and automatically periodizes the NMR signal to be transformed, and therefore the detecting times of the two NMR signals at the input and output of the filter must have the same value to obtain identical periodic times.
This specification refers exclusively to this calculation process RV, as is described in more detail below, first in connection with prior art and subsequently with the inventive method.
In a first step of the conventional calculation process RV which is usually used today for high-resolution NMR spectroscopy (TB<<TA), the filtered NMR signal is multiplied with a window function W1(t) (see FIG. 7, region 3) which is especially provided for this calculation process, wherein an NMR signal of length TA remains (see FIG. 7, region 4). This is achieved by selecting W1(t) such that a region of 2TB at the end of the signal is multiplied by zero and can therefore be cut off. In this manner, an NMR signal of a length TA remains, as desired.
If only the Portion B2 without the portion [B2] of the decaying oscillations is cut-off, this would lead to distortions in the NMR spectrum. To prevent this from happening, both portions B2 as well as [B2] within the NMR signal [F]′ have to be cut off. Elimination of all decaying oscillations in the end region of the NMR signal F′, and not only of parts thereof, eliminates any distorting influence thereof on the NMR spectrum. The fact that signal portions are cut off with this method results in the loss of some information which is very small for high-resolution NMR spectroscopy, since usually TB<<TA.
The cut-off process itself can produce abrupt signal decay at the end of the filtered NMR signal, so that base line distortions in the NMR spectrum are produced if no counter-measures are taken. To prevent this, the filtered NMR signal must subsequently be multiplied with a suitable window function W1(t) thereby smoothing the abrupt decay.
This application of window functions is generally common in NMR, in particular, if there is not sufficient time to permit complete decay of the NMR signal (steady state experiments, multi-dimensional experiments) and if there is still sufficient signal-to-noise ratio and resolution.
In a last step, the steep phase characteristic caused by the digital filter is compensated for using a large linear phase correction, which can be obtained either through corresponding phase correction in the frequency domain or through cyclic rotation of the NMR signal in the time domain. In the second case, the rising oscillation B1″ is shifted to the end of F″ (see FIG. 7, region 5).
The digital NMR signal F′ at the output of the digital filter usually contains undesired disturbing components which are caused by the quality of the receiver electronics used. A disturbing DC component may e.g. exist which must in any event be eliminated before application of the window function W1(t) and Fourier transformation, since additional disturbing components could otherwise be generated in the NMR spectrum.
The illustrations of FIG. 8 show an NMR signal with an exaggerated high DC component SDC. Region 1 shows the DC portion before filtering and region 2 after filtering, both drawings not showing the existing rising and decaying oscillations. Region 3 shows only the DC component itself without the FID, in the present case, however, with the existing rising and decaying oscillations. After cutting a length 2TB off the end of the function F′, a DC component is produced which consists of a step like transition function of a height SDC which is delayed by TB, with rising and decaying oscillations at the transition point (see FIG. 8, region 4).
The value SDC can be easily determined from the function F′ by selecting the end region of the function F′ and determining therefrom the value SDC through forming an average value of the function. Since SDC is now known, the pure step function without rising and decaying oscillations can be uniquely defined. This function, however, is not appropriate for compensating the DC component of F′ since the rising and decaying oscillations of F′ would still remain uncompensated and cause distortions in the NMR spectrum.
The rising and decaying oscillations for a pure step function of a height SDC can be determined through calculation, since all parameters of the digital filter are known. Adding the calculated rising and decaying oscillations to the pure step function produces the desired function for complete compensation of the DC portion of F′.
In imaging MRI (TB ≈TA and TB>TA), the situation can be completely different. Depending on the selected characteristic of the digital filter, the group delay time TB can become approximately equal or larger than TA, such that the calculation process RV described in connection with high-resolution NMR spectroscopy would completely fail in this case. Since MRI usually works with echo signals (see FIG. 9 region 1), which start with a small value, reach a maximum value and subsequently drop again to finally once more assume a low value, a simple solution is still possible.
Region 2 of FIG. 9 shows the echo signal at the output of the digital filter. Obviously this signal meets the condition TB>TA. The rising and decaying oscillations B1 and B2 are usually negligibly small and can therefore be set to zero using the window function W1(t) (see FIG. 9, region 3) and be cut off. Also in this case, an echo signal of a length TA is obtained (see FIG. 9, region 4).
Cutting off the very weak rising and decaying oscillations nevertheless represents a small loss in information which may still noticeably deteriorate the quality of the MRI images.
The echo signal may also have a DC portion which can be compensated for as in high-resolution NMR spectroscopy.
If other MRI measuring methods are used for MRI which utilize e.g. semi-echo signals or FID signals, it must be ensured that TB<<TA which is only possible if the group delay time of the digital filter is correspondingly small. If this is the case, the already described method for high-resolution NMR spectroscopy can be used.
The calculation process RV is composed of the following three steps which must be carried out in sequence and in the stated order:                1. Compensation of the DC portion in the filtered NMR signal F′        2. Multiplication with a suitable window function W1(t)        3. Compensation of the phase characteristic to zero by placing the rising oscillation B1 at the end of the NMR signal (this step is not important in MRI since the rising oscillation has been cut off).        
This calculation process RV has the following disadvantages:                1. The group delay time of the digital filter cannot be freely selected but must be adjusted to the existing NMR or echo signal. This is especially the case for MRI.        2. For compensation of the DC portion, the rising and decaying oscillations must be calculated using the filter parameters with the consequence that the producers of NMR spectrometers must pass on the filter parameters of their FIR filters to foreign application software providers. Secret know how could thereby reach the competitors.        3. The calculation process RV can be carried out only using window functions.        4. In high-resolution NMR spectroscopy, part of the FID signal must be cut off thereby losing part of the information.        5. MRI usually requires working with echo signals. Although this is almost general practice anyway, it still limits the possibilities.        6. In MRI, the rising and decaying oscillations B1 and B2 must be cut off. These are generally very small compared to the echo signal itself. Information is nevertheless lost thereby possibly deteriorating the sharpness of the sectional images.        